The twenty-third post is the September 5, 2012 version in the usual place. The significant new additions deal with local complete intersections, regular sequences, depth, and Cohen-Macaulayness. In particular, the third-last chapter to be public is now out (Chapter 28). (Still to come: formal functions and related notions; and etale/smooth/unramified morphisms.)
As always, I am very interested in comments, corrections, and suggestions, no matter how small. (Except: as usual, I am not yet interested in issues of formatting, figures, indexing, and other things that are best dealt with en masse later.)
The new chapter (Chapter 28) is only 10 pages long, but there are notable changes earlier on. Moreover, you may be surprised that the file is shorter — the last page is now p. 690, not 692. This is all a sign that my understanding of these topics has greatly improved while placing them into the architecture of these notes. (Expressions of gratitude: Long ago, Karen Smith and Mike Roth separately explained how to think about these ideas in a good way. More recently, Burt Totaro helped me clean up my thinking, and pointed me to some good references.) Many of the things I wanted to add were better discussed as part of earlier discussions, rather than waiting so late. (I am conscious that this makes the earlier chapters even longer. I am aware of how foie gras is made, and I do not want to do the same thing to these notes, or to the reader.)
Here is a list of changes, culminating with the new chapter 28 on “Depth and Cohen-Macaulayness”. Some specific suggestions for learners are included, as well as some questions for other readers, especially about whether some exercises are gettable given what I say.
9.4 Regular sequences and locally complete intersections
is a new section in the chapter on closed subschemes. Regular sequences are introduced here. Effective Cartier divisors are now introduced here (rather than being some stray comment wherever they were before), as a first example. The reason for the “local” nature of regular sequences is discussed, and the key hard result is that regular sequences remain regular upon any reordering if the ring is local (Theorem 9.4.4). This section is not hard, although I fear it may be distracting. But I feel happier that local complete intersections have a key place in the exposition. (They did not come up naturally for me when I first learned algebraic geometry, and I paid a price for this.)
13.4 Regular local rings are integral domains
is a new section, whose main purpose is the theorem stated in the title. This is a surprisingly hard fact, and I now use it later several times. My goal (as always) is to get what I need as quickly as possible, rather developing all the surrounding theory. Length (used later) and associated graded rings (not used later) are introduced.
Some interesting points:
In the appendix to Fulton’s Intersection Theory, Lemma A.6.2, gives a delightful short proof using blowing-up; it implicitly works only for varieties. Geometers may like it.
- From the inequality ( is the degree of the Hilbert polynomial of a local ring ), we can get a proof of Krull’s Principal Ideal Theorem. But I need the “multi-equation” version of Krull’s theorem, and I couldn’t see how to prove it in as easy a way, so I’ve kept the proof of Krull’s theorem currently in the notes, which is less motivated, but short and double-starred. But if anyone knows of a short proof of the multi-equation version of Krull’s theorem in this vein, please let me know!
- It is true that , but we don’t need it, so I don’t go through the significantly extra work to show it.
- A question about something earlier in Chapter 13: I am disturbed that I know why a smooth -scheme is nonsingular at closed points only if is perfect. (The statement for general is, for example, in tag 00TT of the stacks project, but I find this a depressingly hard fact.) Does anyone know of an easy explanation?
For learners: please let me know if you can understand this section, and can do the exercises. In particular, 13.4.C, 13.4.G, and 13.4.H are important — can you get them? Exercises 13.4.D and 13.4.E are lots of fun.
The old section 13.3 on “two pleasant [unproved] facts” is now cannibalized, as the theorem that localizations of regular local rings are regular is now stated back in 13.2.14 (and we now prove the case of localizations of finite type algebras over perfect fields, see Thm. 13.2.15 and the Chapter 22 discussion below), and the theorem that regular local rings are UFDs is stated here in 13.4.
- Section 13.7 on completions is now removed, as it is no longer used. (It used to be used to show that regular local rings were integral domains in the case of particularly nice varieties.) The only downside is that I’ve lost the definition of “node”, which at some point will have to be put back in.
- On a related note: I removed the section on flatness and completion (in the flatness chapter), because it never was used, and it allowed me to remove 13.7.
In Chapter 22 (Differentials):
The conormal sheaf to a local complete intersection is easily and quickly shown to be locally free, in Proposition 22.2.16.
The conormal exact sequence is shown to be left-exact for closed embeddings of smooth varieties in Theorem 22.2.26. I was surprised at how easy this was. (Is this similarly easy to learners? Admittedly, it uses a lot of things from earlier on.) I did not do left-exactness in more general situations, but gave references (Remark 22.2.27). It is possible I will do some of this in the forthcoming chapter on smoothness, but I very possibly will not — we don’t need it, and right now it seems like hard work.
The fact that localizations of regular local rings are regular for localizations of finitely generated algebras over perfect fields (Theorem 13.2.15) is proved in 22.4.10 and 22.4.J. This was surprisingly easy to me. (Do you agree? Disagree?) So we finally know that affine space over a perfect field is nonsingular!
Learners: can you get Exercise 22.3.D on differentials of discrete valuation rings? I’ve reworked it, following an excellent suggestion of John Pardon.
Chapter 23: Blowing up
This starred chapter used to be Chapter 19, but is now moved after differentials, because the conormal sheaf/cone/bundle plays an important role in later subsections. Now added: for a local complete intersection, the exceptional divisor is the projectivized normal bundle (and related facts, see Exercise 23.3.D), and the blow-up of a smooth subvariety of a smooth variety is smooth (Theorem 23.3.10). This relies on the fact that for a local complete intersection, is an isomorphism. This requires a little work; I followed Fulton’s slick argument in A.6.1 in Intersection Theory.
We finally come to:
Chapter 28: Depth and Cohen-Macaulayness.
Although Koszul complexes are a central tool for understanding depth and Cohen-Macaulayness, I avoid using them, following my usual philosophy of moving as briskly as possible to what we need, and not developing surrounding theory.
18.1 is introduces depth of Noetherian local rings. Because I find it an algebraic rather than geometric concept, we concentrate on developing some geometric sense of what it means. The main technical result in this section is the argument the cohomological interpretation of maximal sequences.
18.2 introduces Cohen-Macaulayness. A number of results are shown (e.g. equidimensionality, no embedded points, slicing criterion). Important but unneeded results are only stated in 28.2.13. A highlight of this section is the short proof of the “miracle flatness theorem”, which we make important use of in the two last chapters. (It is highly possible that this name is due to Brian Conrad. I like it.)
18.3 gives Serre’s criterion for normality. It seemed worth including, because we would like to know that regular local rings are integrally closed (with proof, not just invoking a fact we haven’t proved), so we can use all of our foundational work on normal schemes. But I’ve starred this section, in the hopes that people will read it only if they really want to.
Question for experts: In 18.3, to show that regular local rings are normal, I need the fact that they are R1. Is there an easy explanation for this fact? Currently, I quote the hard fact that localizaton of regular local rings are local, which I actually prove in the case most interesting to most people (finite type over a perfect field, see Theorem 13.2.15 above).
Please let me know which exercises you find difficult! (And, if you have the time, even which exercises you solved!)
Exercises you should certainly try: 28.1.A, 28.1.F, 28.2.A, 28.2.B, 28.2.D, 28.2.E (the most fun exercise in this chapter), 28.2.F, 28.3.B.
Please tell me how hard you find these, and if you get them: 28.2.D and 28.2.F.